6 NEIGHBORHOODS 35

(3.6.1) For any nonempty set A c E, and any r > 0, the set V/A) =
{x e E | d(x, A) < r} is an open neighborhood of A.

For if d(x, A) < r and d(x, y) < r - d(x, A), it follows from (3.4.2) that
d[y, A) < d(x9 A) + r — d(x, A) = r, hence Vr(A) is open, and obviously
contains A.

When A = {a}, Vr(A) is the open ball B(#; r).

A fundamental system of neighborhoods of A is a family (UA) of neigh-
borhoods of A such that any neighborhood of A contains one of the sets UA.
For arbitrary sets A, the Vr(A) (r > 0) do not in general form a fundamental
system of neighborhoods of A (see however (3.17.11)). It follows from the
definitions that:

(3.6.2) The balls B(a; l/ri) (n integer > 0) form a fundamental system of
neighborhoods of a.

(3.6.3) The intersection of a finite number of neighborhoods of A. is a neigh-
borhood of A.

This follows from (3.5.3).

(3.6.4) In order that a set A be a neighborhood of every one of its points., a
necessary and sufficient condition is that A be open.

The condition is obviously sufficient; conversely, if A is a neighborhood
of every x e A, there exists for each x e A an open set U* c A which contains
x. From the relations x e Ux c A we deduce A = (J {x} <= (J U^ c A,

xeA xeA

hence A = (J U* is an open set, by (3.5.2).

PROBLEM

In the real line, show that the subset N of all integers ^ 0 does not possess a denumer-
able fundamental system of neighborhoods. (Use contradiction, and apply the following
remark: if (amn) is a double sequence of numbers > 0, the sequence (£„) where bn = am/2 is
such that for no integer m does the inequality bn ^ amn hold for all integers n.) t*ec U); the I' i%oes not belong to a ball B(a; r) (resp. B'(#; r)), then